Re: Wilcoxon Mann-Whitney Rank Sum Test in R

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

Bob Green


An earlier post had posed the question: "Does anybody know what is relation
between 'T' value calculated by 'wilcox_test' function (coin package) and
more common 'W' value?"

I found the question interesting and ran the commands in R and SPSS. The W
reported by R did not seem to correspond to either   Mann-Whitney U,
Wilcoxon W or the Z which I have more commonly used. Correction for ties
may have affected my results.

Can anyone else explain what the reported W is and the relation to the
reported T?

regards

bob

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

Peter Dalgaard
Bob Green <[hidden email]> writes:

> An earlier post had posed the question: "Does anybody know what is relation
> between 'T' value calculated by 'wilcox_test' function (coin package) and
> more common 'W' value?"
>
> I found the question interesting and ran the commands in R and SPSS. The W
> reported by R did not seem to correspond to either   Mann-Whitney U,
> Wilcoxon W or the Z which I have more commonly used. Correction for ties
> may have affected my results.
>
> Can anyone else explain what the reported W is and the relation to the
> reported T?

Well, it's open source... You could just go check.

W is the sum of the ranks in the first group, minus the minimum value
it can attain, namely sum(1:n1) == n1*(n1+1)/2. In the tied cases, the
actual minimum could be larger.

The T would seem to be asymptotically normal

> wilcox_test(pd ~ age, data = water_transfer,distribution="asymp")

        Asymptotic Wilcoxon Mann-Whitney Rank Sum Test

data:  pd by groups 12-26 Weeks, At term
T = -1.2247, p-value = 0.2207
alternative hypothesis: true mu is not equal to 0

> pnorm(-1.2247)*2
[1] 0.2206883

so a good guess at its definition is that it is obtained from W or one
of the others by subtracting the mean and dividing with the SD.

--
   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - ([hidden email])                  FAX: (+45) 35327907

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

P Ehlers


Peter Dalgaard wrote:

> Bob Green <[hidden email]> writes:
>
>
>>An earlier post had posed the question: "Does anybody know what is relation
>>between 'T' value calculated by 'wilcox_test' function (coin package) and
>>more common 'W' value?"
>>
>>I found the question interesting and ran the commands in R and SPSS. The W
>>reported by R did not seem to correspond to either   Mann-Whitney U,
>>Wilcoxon W or the Z which I have more commonly used. Correction for ties
>>may have affected my results.
>>
>>Can anyone else explain what the reported W is and the relation to the
>>reported T?
>
>
> Well, it's open source... You could just go check.
>
> W is the sum of the ranks in the first group, minus the minimum value
> it can attain, namely sum(1:n1) == n1*(n1+1)/2. In the tied cases, the
> actual minimum could be larger.
>
> The T would seem to be asymptotically normal
>
>
>>wilcox_test(pd ~ age, data = water_transfer,distribution="asymp")
>
>
>         Asymptotic Wilcoxon Mann-Whitney Rank Sum Test
>
> data:  pd by groups 12-26 Weeks, At term
> T = -1.2247, p-value = 0.2207
> alternative hypothesis: true mu is not equal to 0
>
>
>>pnorm(-1.2247)*2
>
> [1] 0.2206883
>
> so a good guess at its definition is that it is obtained from W or one
> of the others by subtracting the mean and dividing with the SD.
>

With the SD adjusted for ties, of course. (See, e.g., Conover's book.)

Peter Ehlers
University of Calgary

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

Peter Dalgaard
P Ehlers <[hidden email]> writes:

> > so a good guess at its definition is that it is obtained from W or one
> > of the others by subtracting the mean and dividing with the SD.
> >
>
> With the SD adjusted for ties, of course. (See, e.g., Conover's book.)

...which is actually the exact SD, conditional on the set of tied
ranks, not just a correction term. See my discussion with Torsten a
month or so ago.

--
   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - ([hidden email])                  FAX: (+45) 35327907

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

Torsten Hothorn

On Wed, 21 Dec 2005, Peter Dalgaard wrote:

> P Ehlers <[hidden email]> writes:
>
> > > so a good guess at its definition is that it is obtained from W or one
> > > of the others by subtracting the mean and dividing with the SD.
> > >
> >
> > With the SD adjusted for ties, of course. (See, e.g., Conover's book.)
>
> ...which is actually the exact SD, conditional on the set of tied
> ranks, not just a correction term. See my discussion with Torsten a
> month or so ago.
>

yes, exactly. Thanks, Peter!

The `T' values reported by functions in the `coin' package are
_standardized_ statistics. Standardization is done utilizing the
conditional expectation and conditional variance of the underlying linear
statistics as given by Strasser & Weber (1999). Note that _no_
`continuity correction' whatsoever is applied. The limit distribution is
normal (or chisq, when the test statistic is a quadratic form).

The vignette explains the theoretical framework `coin' maps into
software in more detail. It _definitively_ is worse the effort to
have a look at it. At first glance it might seem a little bit
abstract but after this you'll see how general and powerful the
tools are.

We are currently working on a manuscript showing more applications, so
watch out for the new `coin' version in a few days.

Best,

Torsten

> --
>    O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
>   c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
>  (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
> ~~~~~~~~~~ - ([hidden email])                  FAX: (+45) 35327907
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
>
>

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Re: Wilcoxon Mann-Whitney Rank Sum Test in R

P Ehlers
In reply to this post by Peter Dalgaard
Peter,

You're right, of course, as usual. Sorry about that.

Peter E.


Peter Dalgaard wrote:

> P Ehlers <[hidden email]> writes:
>
>
>>>so a good guess at its definition is that it is obtained from W or one
>>>of the others by subtracting the mean and dividing with the SD.
>>>
>>
>>With the SD adjusted for ties, of course. (See, e.g., Conover's book.)
>
>
> ...which is actually the exact SD, conditional on the set of tied
> ranks, not just a correction term. See my discussion with Torsten a
> month or so ago.
>

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